Spectral gaps and oscillations Alexei Poltoratski Texas A&M Abel Symposium, 8/2012 Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 1 / 22
Contents of the talk Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 2 / 22
Contents of the talk The Gap Problem: Estimating the size of the gap in the Fourier spectrum of a measure. Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 2 / 22
Contents of the talk The Gap Problem: Estimating the size of the gap in the Fourier spectrum of a measure. The Type Problem: Completeness of complex exponentials in L 2 -spaces. Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 2 / 22
Contents of the talk The Gap Problem: Estimating the size of the gap in the Fourier spectrum of a measure. The Type Problem: Completeness of complex exponentials in L 2 -spaces. A Problem on Oscillations of Fourier Integrals: How often should a measure with a spectral gap change signs? Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 2 / 22
Beurling’s Gap Problem Let X ⊂ R be a closed set. Question: Under what conditions on X does there exist a non-zero finite complex measure µ, supp µ ⊂ X whose Fourier transform � e 2 π ixt d µ ( t ) µ ( x ) = ˆ vanishes on an interval? How to determine the maximal size of such an interval (spectral gap)? The Gap Problem is a part of an area called Uncertainty Principle in Harmonic Analysis. In this context the principle says that the supports of the measure and its Fourier transform cannot both be small. Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 3 / 22
Beurling’s Gap Problem Definition If X is a closed subset of the real line denote G X = sup { a | ∃ µ �≡ 0 , supp µ ⊂ X , such that ˆ µ = 0 on [0 , a ] } Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 4 / 22
Beurling’s Gap Problem Definition If X is a closed subset of the real line denote G X = sup { a | ∃ µ �≡ 0 , supp µ ⊂ X , such that ˆ µ = 0 on [0 , a ] } Examples: 1) If X is bounded from below or from above then G X = 0 . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 4 / 22
Beurling’s Gap Problem Definition If X is a closed subset of the real line denote G X = sup { a | ∃ µ �≡ 0 , supp µ ⊂ X , such that ˆ µ = 0 on [0 , a ] } Examples: 1) If X is bounded from below or from above then G X = 0 . 2) If µ is the counting measure of Z then ˆ µ = µ in the sense of distributions (Poisson formula). It follows that G 1 d Z = d . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 4 / 22
Beurling’s Gap Problem Definition If X is a closed subset of the real line denote G X = sup { a | ∃ µ �≡ 0 , supp µ ⊂ X , such that ˆ µ = 0 on [0 , a ] } Examples: 1) If X is bounded from below or from above then G X = 0 . 2) If µ is the counting measure of Z then ˆ µ = µ in the sense of distributions (Poisson formula). It follows that G 1 d Z = d . 3) Since Y ⊂ X ⇒ G Y ≤ G X , If X contains 1 d Z + c then G X ≥ d . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 4 / 22
Beurling’s Gap Theorem Definition A sequence of disjoint intervals { I n } is long if | I n | 2 � 1 + dist 2 (0 , I n ) = ∞ and short otherwise. Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 5 / 22
Beurling’s Gap Theorem Definition A sequence of disjoint intervals { I n } is long if | I n | 2 � 1 + dist 2 (0 , I n ) = ∞ and short otherwise. Theorem (Beurling’s Gap Theorem) If the complement of X is long then G X = 0 . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 5 / 22
A solution to the Gap Problem: Energy Let Λ = { λ 1 , ..., λ n } be a finite set of points on R . Consider the quantity � L (Λ) = log | λ k − λ l | . k � = l Physical interpretation: L (Λ) is the energy of a system of electrons placed at the points of Λ (2D Coulomb gas). Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 6 / 22
A solution to the Gap Problem: Energy Let Λ = { λ 1 , ..., λ n } be a finite set of points on R . Consider the quantity � L (Λ) = log | λ k − λ l | . k � = l Physical interpretation: L (Λ) is the energy of a system of electrons placed at the points of Λ (2D Coulomb gas). Key example: Let I ⊂ R be an interval, Λ = I ∩ 1 D Z . Then L (Λ) = ∆ 2 log | I | + O ( | I | 2 ) ∆ = #Λ = D | I | + o ( | I | ) , as follows from Stirling’s formula. (Note: max L (Λ) is attained when the electrons are placed at the endpoints of the interval and at the roots of the Jacobi polynomial of degree k − 2.) Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 6 / 22
A solution to the Gap Problem: Short partitions Let ... < a − 2 < a − 1 < a 0 = 0 < a 1 < a 2 < ... be a two-sided sequence of real points. We say that the intervals I n = ( a n , a n +1 ] form a short partition of R if | I n | → ∞ as | n | → ∞ and the sequence { I n } is short, i.e. | I n | 2 � 1 + dist 2 ( I n , 0) < ∞ . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 7 / 22
A solution to the Gap Problem: D -uniform sequences Let Λ = { λ n } be a sequence of distinct real points. We say that Λ is D -uniform if if there exists a short partition I n such that ∆ n = D | I n | + o ( | I n | ) as n → ±∞ (density condition) and ∆ 2 n log | I n | − L n � 1 + dist 2 (0 , I n ) < ∞ (energy condition) n where � ∆ n = #(Λ ∩ I n ) and L n = L (Λ ∩ I n ) = log | λ k − λ l | . λ k ,λ l ∈ I n , λ k � = λ l Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 8 / 22
A solution to the Gap Problem: The main theorem Recall G X = sup { a | ∃ µ �≡ 0 , supp µ ⊂ X , such that ˆ µ = 0 on [0 , a ] } Theorem G X = sup { D | X contains a D -uniform sequence } . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 9 / 22
Corollaries: separated sequences Interior BM density of a discrete sequence: D ∗ (Λ) = inf { d | ∃ long { I n } such that #(Λ ∩ I n ) � d | I n | , ∀ n } . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 10 / 22
Corollaries: separated sequences Interior BM density of a discrete sequence: D ∗ (Λ) = inf { d | ∃ long { I n } such that #(Λ ∩ I n ) � d | I n | , ∀ n } . Λ is separated if | λ n − λ k | > c > 0 for all n � = k . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 10 / 22
Corollaries: separated sequences Interior BM density of a discrete sequence: D ∗ (Λ) = inf { d | ∃ long { I n } such that #(Λ ∩ I n ) � d | I n | , ∀ n } . Λ is separated if | λ n − λ k | > c > 0 for all n � = k . Corollary (M. Mitkovski, A.P.) If Λ ⊂ R is a separated sequence then G Λ = D ∗ (Λ) . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 10 / 22
Corollaries: separated sequences Interior BM density of a discrete sequence: D ∗ (Λ) = inf { d | ∃ long { I n } such that #(Λ ∩ I n ) � d | I n | , ∀ n } . Λ is separated if | λ n − λ k | > c > 0 for all n � = k . Corollary (M. Mitkovski, A.P.) If Λ ⊂ R is a separated sequence then G Λ = D ∗ (Λ) . Example 1 2 − ε ). Then Let Λ = { λ n } be a separated sequence such that λ n − n = O ( n G Λ = 1 . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 10 / 22
The Type Problem Let µ be a finite positive measure on the real line. For a > 0 denote by E a the family of exponential functions E a = { e 2 π ist | s ∈ [0 , a ] } . The exponential type of µ : T µ = inf { a > 0 | E a is complete in L 2 ( µ ) } if the set of such a is non-empty and infinity otherwise. Problem Find T µ in terms of µ . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 11 / 22
The Type Problem: History This question first appears in the work of Wiener, Kolmogorov and Krein in the context of stationary Gaussian processes in 1930-40’s. If µ is a spectral measure of a stationary Gaussian process, the property that E a is complete in L 2 ( µ ) is equivalent to the property that the process at any time can be predicted from the data for the time period from 0 to a . The type problem can also be restated in terms of the Bernstein weighted approximation, see for instance Koosis’ book. Important connections with spectral theory of second order differential operators were studied by Gelfand, Levitan and Krein. Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 12 / 22
Known results A classical result by Krein (1945) says that if d µ = w ( x ) dx and log w ( x ) / (1 + x 2 ) is summable then T µ = ∞ . A partial inverse, proved by Levinson and McKean (1964), holds for even monotone w . Alexei Poltoratski (Texas A&M) Spectral gaps and oscillations Abel Symposium, 8/2012 13 / 22
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